At the end of the 1970s, researchers at the Institute for Research on Mathematics Teaching (IREM) in Grenoble, France, gave primary-school children a strange problem:

There are 26 sheep and 10 goats on a ship. How old is the captain?

The problem has no answer: nothing in it mentions anyone’s age. Even so, according to the account that the educator Stella Baruk made famous in her book L’âge du capitaine (1985), 76 out of 97 children answered with a number, almost always $36$, the sum of the two that appeared in the text.

The hidden contract

The children were not absent-minded. They were following, quite logically, the rules school had taught them without saying so: every problem has an answer, the answer is a number, every number in the statement must be used, and the operation is the one the words suggest. The French researcher Guy Brousseau called this set of unspoken rules the didactic contract. When the contract matters more than the text, the text stops being read.

The key words that mislead

One of the rules of the contract is that of key words: “more” means add, “left” means subtract, “each” means multiply. It works in many textbook exercises, and that is why it sticks. But it is enough to turn the sentence around:

Anna has 12 sweets, 4 more than Bea. How many sweets does Bea have?

Whoever obeys the word “more” adds and answers $16$. Whoever reads notices that Bea has fewer: $12 - 4 = 8$. Written in the register of signs, the statement is $A = B + 4$, with $A = 12$; the equation has a “$+$”, and yet the calculation required is a subtraction. The word describes the relation between the quantities, not the operation the reader should perform.

This passage from text to equation is a conversion between registers, in Duval’s sense, which we saw in chapter 2. It is hard precisely when the order of the words does not match the order of the signs.

Pólya’s four steps

The Hungarian mathematician George Pólya summed up, in How to Solve It (1945), a plan that is still useful:

  1. Understand the problem. What is the unknown? What are the data? What is the condition? Can you state the problem in your own words?
  2. Devise a plan. Do you know a related problem? Can you name the quantities and write the condition as an equation?
  3. Carry out the plan. Do the calculations, checking each step.
  4. Look back. Does the answer make sense? Does it have the right unit? Does it answer the question that was asked?

It is foolish to answer a question that you do not understand.

— George Pólya, How to Solve It (1945)

The first step would have saved the Grenoble children: the unknown is a person’s age, and no datum mentions it. The last one too: a $36$-year-old captain is plausible, but the question “where did this $36$ come from?” has no answer.

Exercise: the discount in reverse

After a $20\%$ discount, a shirt costs 40 reais. What was the original price? Why is the answer “48 reais” wrong?

Calling the original price $p$, the statement says $0.8\,p = 40$, so $p = 50$. The answer $48$ comes from the key word: “20% discount” suggests working with $20\%$ of $40$. But the $20\%$ is of the original price, not of the final one. Checking: $20\%$ of $50$ is $10$, and $50 - 10 = 40$.

Exercise: problems without an answer

Which of the two problems can be solved? (a) “A class has 30 students and 18 are girls. How many boys are older than 10?” (b) “A class has 30 students and 18 are girls. How many boys are there?”

Only (b): $30 - 18 = 12$ boys. In (a), the data say nothing about ages, and any number would be made up. A good classroom exercise is precisely to mix problems with and without answers and ask students to say, before calculating, which ones can be solved.

Chapter index · ← Two languages in one sentence: the mathematics register · The conventions nobody writes down →